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I am indeed claiming that it works over any field, but with the additional assumption that the morphism f be separable (this is required to make the induced morphism between the normalizations an isom …

The argument that I had is flawed and I do not know how to fix it. Here is a short description of one part that I cannot address. We are given two polarized K3 surfaces X and Y of degrees x and y resp …

I believe that as you stated it, an affinization cannot exist. Consider $\mathbb{P}^2$ with a point removed p. Any line through p is affine and should "survive" affinization. Every curve not through p …

I think that the quartic with equation \[ x^4 + y^4 + xy^2z + yw^3 + z^3w \] over $\mathbb{F}_2$ has no line and that it has no conic defined over a field of size at most $2^8$. Both statements I hav …

A few opposite-looking remarks, and the case of (minimal) surfaces. If a variety $X$ admits a non-constant morphism to a curve, then it admits a non-constant morphism to $\mathbb{P}^1$ (just compose …

Certainly not: even in the case of $X=Y=S=\mathbb{P}^1$ and the two maps $X,Y \to S$ are the same and general of degree at least three. In this case one component is the "diagonal" $\mathbb{P}^1$ and …

Let me expand on my comment. Let $E$ be an elliptic curve and let $p$ be a point of $E$ of infinite order. Embed $E$ in $\mathbb{P}^2$ as a plane cubic using the linear system $3O$, where $O$ is the …

This is a substantial revision of my previous answer based on the comments and the other answers. I am making it community wiki for two reasons: it incorporates ideas from my answer (for which I alrea …

I do not know about the local-to-global principle for testing rationality of a surface, but weakening "rational" to "unirational" it certainly fails. Indeed, del Pezzo surfaces violating the Hasse pr …

I would like to mention one more homotopy invariant of smooth projective varieties that is also a birational invariant. If $X$ is a smooth projective variety, then the torsion subgroup $T(X)$ of ${\rm …

A result of Kawamata (Kawamata, Yujiro, Characterization of abelian varieties. Compositio Mathematica, 43 no. 2 (1981), p. 253-276) implies that, under your assumptions, $X$ is birational to an abelia …

Let $\ell_1,\ell_2,\ell_3$ be three lines in the plane that do not all contain the same point. The triangle formed by $\ell_1,\ell_2,\ell_3$ is the set obtained from $\ell_1 \cup \ell_2 \cup \ell_3$ …

This is mostly a series of comments, but guided by the questions you asked. First of all, I will only talk about $X_n$, interpreting it as the space of non-zero complex polynomials $p$ of degree at m …

I see that Sándor already gave an answer to the question, but I wanted to expand on my comment, giving an answer that does not rely on the Basepoint-free Theorem. It exploits the fact that ampleness …

I do not know how much progress has been made on this, but what you ask is part of a conjecture appearing in a paper of Eisenbud, Green and Harris (see Cayley-Bacharach Theorems and Conjectures, Conje …